What are the coordinates of point F and point D?

Question

AD:AB= 2:5, the coordinates of point F are A (4.4, -4.6)

                                                               B (4.4, -3.6)

                                                               C (4.6, -4.6)

                                                               D (4.6, -3.6)

 

Point E has the coordinates (2.8, -3), and the coordinates of point D are,

A (4.4, -2.4)

B (4.4, -1.4)

C (4.6, -2.4)

D (4.6, -1.4)

 

Answer 1

Because you were given segment ratios, we will use that information to solve.  I will offer other approaches in the end.  Looking at your figure, we can conclude that...

AD≅AF≅FE≅DE

AD:AB=2:5....given

Pt B=(10,1)...given

Pt A=(6,-3)....given

 

 

Let's look at AB being the hypotenuse of a right triangle with sides x₁ and y₁.  Sketch these legs dashed on your figure, label legs accordingly.  From the coordinates given, we can calculate x₁ (horizontal leg) as...

x₁=B_x-A_x=10-6=4.....x-coordinates used, Pts A and B

 

...and calculate y₁ (vertical leg) as...

y₁=B_y-A_y=1-(-3)=4...y-coordinates used, Pts A and B

 

Therefore, the triangle with AB as the hypotenuse is a 45-45-90 degree right triangle, legs x₁ and y₁ are equal.

 

 

Let's look at AF being the hypotenuse of a right triangle  with sides x₂ and y₂.  Sketch these legs dashed on your figure, label legs accordingly. We need to calculate these values.  We have similar triangles with hypotenuses AF and AB.  All sides of similar triangles are in proportion.  Setting up those proportions using the given ratio...

 

AB/x₁=AF/x₂

   5/4=2/x₂.......substitute ratio values for 

                       "AB and AF, x₁"

   5x₂=2(4).......cross-multiply

   5x₂=8...........simplify

     x₂=8/5........divide both sides by 5

     x₂=1.6........horizontal leg

 

Take the x-coordinate from Pt A and subtract "x₂"...

6-1.6=4.4.....x-coordinate for Pt F.

 

Because we have a 45-45-90 degree right triangle...

y₂=x₂=1.6

 

Take the y-coordinate from Pt A and subtract "y₂"...

-3-1.6=-4.6...y-coordinate for Pt F.

 

The coordinates for Point F are (4.4,-4.6).  The correct choice is "A".

 

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I HAVE SET YOU UP WITH A PROCEDURE FOR FINDING ONE OF THE TWO POINTS WITH THE SIDE RATIO GIVEN.  USING THE SAME STEP-BY-STEP PROCEDURE, TRY AND FIND THE COORDINATES FOR POINT "D".  CONSIDER THE SEGMENTS "AD" AND "AF" BEING HYPOTENUSES OF CONGRUENT TRIANGLES, FIND THE LEGS "x₃" AND "y₃" AND CALCULATE THE CORRESPONDING COORDINATE VALUES FOR POINT "D".

 

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Updated edit:

 

Let's examine an alternate solution approach (graphic solution).  You will determine the equations of lines AB and AD...

 

1.  Determine the slope of AB using "m_AB=(y2-y1)/(x2-x1) using the coordinates for Points A and B.

 

2.  Write the equation for line AB using the "point-slope" ((y-y1)=m(x-x1)) form.  Use the value of slope from #1 above and the coordinate values of either Point A or B.

 

3.  Check Point F choices "a,b,c,d" for truth by substituting into the equation for line AB determined in #2 above.  Select the choice that produces a truth.

 

4.  Calculate the slope of segment AD.  Segment AD is perpendicular to segment AB.  Recall the slope of a perpendicular line is "(m_⊥)_AD=-(1/m_AB).  "m_AB" was calculated in #1 above.  

 

5.  Write the equation for line AD using the "point-slope" ((y-y1)=m(x-x1)) form. Use the value of slope from #4 above and the coordinate values of Point A.

 

6.  Check Point D choices "a,b,c,d" for truth by substituting into the equation for line AD determined in #5 above.  Select the choice that produces a truth.

 

 

You can view the graphic solution at the following URL...

 

https://www.wyzant.com/resources/files/428698/linear_graphing

 

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Still another approach is calculate the length of AB using the distance formula "d=√[(x2-x1)²+(y2-y1)²]".  Use this value and set up a length ratio between AB and AF to find length of AF.  AF is equal to AD.  Calculate the legs of the corresponding triangles.  Use the triangle leg lengths to calculate the coordinate values of Points F and D (similar to what I described in the first solution).

 

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Let me know how you have progressed by commenting or emailing a message to me (tap my picture ID badge at the top).  I gladly will follow-up with further explanation if needed.